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TitleStudy on the promotion impact of demand response ondistributed PV penetration by using non-cooperative gametheoretical analysis
Author(s) Wang, Ge; Zhang, Qi; Li, Hailong; McLellan, Benjamin C.;Chen, Siyuan; Li, Yan; Tian, Yulu
Citation Applied Energy (2017), 185(2): 1869-1878
Issue Date 2017-01-01
URL http://hdl.handle.net/2433/246271
Right
© 2017. This manuscript version is made available under theCC-BY-NC-ND 4.0 licensehttp://creativecommons.org/licenses/by-nc-nd/4.0/.; この論文は出版社版でありません。引用の際には出版社版をご確認ご利用ください。; This is not the published version.Please cite only the published version.
Type Journal Article
Textversion author
Kyoto University
1
Study on the Promotion Impact of RTP-based Demand 1
Response on Distributed PV Penetration by using a Non-2
Cooperative Game Theoretical Analysis 3
4
GE WANG 5 Academy of Chinese Energy Strategy, 6 China University of Petroleum at Beijing 7 8 9 QI ZHANG (CORRESPONDING AUTHOR) 10 Academy of Chinese Energy Strategy, 11 China University of Petroleum-Beijing, Changping, Beijing 102249, China 12 Email: zhangqi@cup.edu.cn; zhangqi56@tsinghua.org.cn 13 14 Benjamin C. McLellan 15 Graduate School of Energy Science 16 Kyoto University, Japan 17 18 HAILONG LI 19 School of Business, Society and Engineering, 20 Mälardalen University, Sweden 21 22 Yan LI 23 Academy of Chinese Energy Strategy, 24 China University of Petroleum at Beijing 25 26 27 SIYUAN CHEN 28 Academy of Chinese Energy Strategy, 29 China University of Petroleum at Beijing 30 31 32
2
Abstract 33
Promoting the penetration of distributed photovolataic systems (PV) at the end-34
user side is an important contribution to carbon reduction. This study aims to evaluate 35
the promotion impact of the level of smart consumers on the installation of distributed 36
PV using a non-cooperative game theoretical model, which can find the Nash 37
equilibrium of residential smart consumers with different levels of demand control 38
capability in a electricity power market with real-time pricing mechanism under 39
different installed PV capacities and battery capacities. As a case study, 5 levels of smart 40
control, 32 combinations of PV installed capacities and battery capacities were 41
analyzed and inter-compared using the developed model. The results show that: (i) the 42
consumers with higher smart control level are able to accept larger PV capacity because 43
the marginal revenue of new installed PV for smart consumers decreases much more 44
slowly compared to that of a common consumer; (ii) the smarter consumers need less 45
batteries to promote PV economic acceptability; (iii) the smarter consumers can meet 46
the electricity demand in real-time with least expenditure thanks to their advanced 47
demand-response capability, so they get more ultimate benefit from the games. 48
49
Key words: Demand response, Distributed PV, Real-time Pricing, Non-cooperative 50
Game, Complementarity Model 51
3
Nomenclature 52
Indices 53
t hour series in a day (1–24) 54
d representative days (3 typical days) 55
i,j user group (1-5) 56
c controllable appliances 57
58
Parameters 59
NCL(i,d,t) non-controllable load (kW) 60
STA(i,d,c) earliest start time of one controllable appliance 61
STO(i,d,c) latest stop time of one controllable appliance 62
RP(i,d,c) rated power of one controllable appliance (kWh) 63
IRR(d,t) solar irradiation in terms of output power/rated power (%) 64
DAYS(d) amount of days represented by representative day in each year 65
PV(i) installed PV capacity (kWh) 66
BA(i) installed battery capacity (kWh) 67
α, β coefficient of electricity price 68
NRL(d,t) non-residential load (kWh) 69
CHAEFF battery charge efficiency 70
DISEFF battery discharge efficiency 71
72
73
4
Variables 74
be(i,d,t) power bought from grid (kWh) 75
se(i,d,t) power sold to grid (kWh) 76
ca(i,d,c,t) power consumed by one controllable appliance (kWh) 77
pr(d,t) power price established by power retail (RMB per kWh) 78
brin(i,d,t) battery charge rate (%) 79
brout(i,d,t) battery discharge rate (%) 80
81
Abbreviations 82
PV photovoltaic 83
FIT feed-in tariff 84
RTP real-time pricing 85
TOU time of use 86
GAMS General Algebraic Modeling System 87
MCP mixed complementarity problem 88
89
5
1. Introduction 90
With the dramatic increase of fossil fuel consumption, the reduction of CO2 91
emission and the promotion of renewable energy have become urgent tasks for societies. 92
PV, as one of the most important renewable energy technologies, has been commonly 93
used in a distributed configuration, installed very near or at the end user`s location. To 94
promote the application of PV, many kinds of subsidization policies have been proposed 95
[1-3]. For example, in many countries, governments provide a lump-sum grant to 96
consumers who install PV systems or subsidies on the electricity generated by PV. 97
Moreover, utility companies are often obligated to purchase PV power at a price 98
relatively higher than the regular tariff under government-supported feed-in tariffs 99
(FIT). However, in the traditional power grid, PV power is difficult to integrate due to 100
its intermittence and low voltage [4]. Thanks to the recent rapid development of 101
communication and automation technologies using telemetry, remote and automated 102
control have enabled smart grid and demand response, which is expected to help 103
dispatch and utilize PV power in both macro-grids [5, 6] and micro-grids [7, 8]. 104
Apart from policy and subsidies, highly varied electricity prices resulting from the 105
restructuring of the electricity market can also create an incentive to incorporate more 106
PV panels [9]. Practically, energy demand is not constant, but varies from moment to 107
moment depending on end-users patterns of usage. In order to meet the energy demand 108
of consumers, the generation and transmission capacities of the grid need to be designed 109
6
to satisfy the peak power demand rather than the average power demand, leading to an 110
over-capacity system for much of the time. In the traditional power market, the 111
electricity price is usually fixed. A fixed price cannot reflect the fluctuation of 112
generation cost caused by peak load, unit commitment constraints, congested 113
transmission lines etc., and thus consumers have no motivation to shift consumption 114
during the peak load periods. This results in a redundancy of power generation capacity 115
and transmission infrastructures in the off-peak times, and therefore, the whole system 116
becomes inefficient and cost-ineffective. To solve this problem, economic dispatch 117
based on a real-time pricing (RTP) system and demand-response technologies is 118
proposed, which can motivate consumers to shift their loads from peak times. Various 119
pricing strategies have been proposed to incentivize demand-response in smart grid, 120
and the most efficient one is the RTP [10, 11]. Demand response programs under real-121
time pricing markets have been widely adopted in practice, and the promotion impacts 122
on market access to distributed energy have been determined in previous studies [12]. 123
There have been several models developed with a focus on the relationship 124
between demand-response and the installation capacity of distributed PV. The existing 125
models can mainly be divided into two types in terms of the mathematical methodology. 126
One type is the optimal model, including static optimization models solving optimal 127
load commitment problems of electric appliances in smart homes with distributed PV 128
installed [13-15], and dynamic optimization models solving optimal expansion 129
planning problems for PV in smart homes [16, 17]. In these optimal models, the power 130
7
price is usually set as exogenous. The consumers are price-takers rather than players in 131
the market, which means that the consumers’ behavior has no impact on the electricity 132
price. The other type of model is the game theoretical model, for example the non-133
cooperative game models handling games among residential consumers equipped with 134
distributed electricity generators [18-20], Stackelberg game models dealing with games 135
between utility companies and smart end-users (such as residential smart homes) in 136
demand response programs [21], and market equilibrium models focusing on the whole 137
power market [22]. In these game theoretical models, consumers’ strategies can affect 138
the power price. Compared with optimal models, game theoretical models have 139
advantages in handling situations where market participants have different or even 140
conflicting objectives, which is more realistic. However, in existing studies, the 141
differences in the smart levels of consumers, especially the difference between smart 142
consumers and non-smart consumers were not yet considered. In addition, the PV 143
installed capacities are usually exogenously given or obtained as optimal results, which 144
do not indicate the detailed impacts of the level of smart control on PV installation 145
capacities. 146
The purpose of this study is to uncover the impact of the level of smart control on 147
distributed PV installation. In order to analyze the impacts of end users’ participation 148
and different smart levels of consumers in an electricity power market, a non-149
cooperative game theoretical model was developed. In the model, consumers 150
participate in the game in a real-time pricing market and every consumer pursues his/her 151
8
own minimized expenditure on electricity consumption. Different from existing studies, 152
different levels of smart control (smart levels) have been considered in the proposed 153
model to analyze their impact on the integration of distributed PV. The smart level is 154
measured by the ability to respond to price fluctuation. The development of smart 155
technologies is the foundation of the actualization of consumers’ demand-response. 156
Such technologies include smart metering, remote control, and automated control, and 157
so on [10, 23]. Newly built houses are considered to be equipped with the latest 158
commercially-effective advanced smart electric devices and consumers can therefore 159
respond to the price fluctuations quickly and flexibly. One the other hand, old houses 160
are typically not smart enough and thus cannot allocate energy consumption according 161
to price fluctuations. Consumers with different smart levels were then analyzed and 162
compared. The equilibrium results with different installed capacities of PV as well as 163
batteries were obtained. Every consumer’s optimal operation pattern and total expense 164
can be clarified. The economical acceptability of PV was then evaluated, and the 165
feasibility of the methodology was demonstrated practically through its application to 166
a case study in China. 167
2. Methodology 168
2.1. Model Framework 169
The model framework is shown in Figure 1. Non-residential consumers, such as 170
commercial and industrial consumers, usually have contracts with power retailers and 171
9
they are charged a rate based on the time of use (TOU) of their energy. Their electricity 172
price is pre-given, and thus their consumption can be regarded as constant in the real-173
time pricing market. On the other hand, residential consumers are bidirectionally 174
connected to the retailer. The power retailers gather consumption data and set the real-175
time price based on the total wholesale hourly power consumption. The residential 176
consumers respond to the hourly price by shifting their power load in the form of 177
submitting new demand bids and the retailer sets a new price again. This procedure will 178
be repeated until equilibrium is achieved. 179
2.2. Residential micro-grid module 180
Residential micro-grids consist of controllable and non-controllable electric 181
appliances, distributed photovoltaic panels, batteries, and central control appliances. 182
Controllable appliances refer to appliances which can be operated at any time of the 183
day or within a particular time interval while non-controllable appliances refer to those 184
whose operation time is fixed. For example, a washing machine is controllable, 185
however, an air-conditioner is usually non-controllable. Distributed photovoltaic panels 186
are connected to the power grid, and surplus generated power can be sold to the grid at 187
the price of the feed-in tariff. 188
Using automated central control technology, residential consumers can gain real-189
time access to market prices and the operation of controllable electric appliances can 190
be arranged automatically to avoid purchasing power during high price periods. It is 191
important to note that this type of system requires expensive technologies that may not 192
10
be available to all consumers. Alternative solutions such as online real-time power price 193
access and remote control of smart appliances can substitute to some extent. Due to the 194
potential disparities in control technologies and capabilities, it is reasonable to divide 195
the residential consumers into several types according to their ability to response to 196
price fluctuation, in other words their smart level. In this paper, the smart level is 197
measured by length of the time interval in which energy consumption can be shifted. 198
In the residential micro-grid, the primary constraint is to make supply and demand 199
meet. 200
201
𝐼𝐼𝐼𝐼𝐼𝐼(𝑑𝑑, 𝑡𝑡) × 𝑃𝑃𝑃𝑃(𝑖𝑖) + 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) − 𝑠𝑠𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) = ∑ 𝑐𝑐𝑐𝑐(𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡)𝑐𝑐 + 𝑁𝑁𝑁𝑁𝑁𝑁(𝑖𝑖,𝑑𝑑, 𝑡𝑡) +202
𝐵𝐵𝐵𝐵(𝑖𝑖) × 𝑁𝑁𝐶𝐶𝐵𝐵𝐶𝐶𝐶𝐶𝐶𝐶 × 𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) − 𝐵𝐵𝐵𝐵(𝑖𝑖) × 𝐷𝐷𝐼𝐼𝐷𝐷𝐶𝐶𝐶𝐶𝐶𝐶 ×203
𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖, 𝑑𝑑, 𝑡𝑡), ( 𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡)) ∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 204
(2-1) 205
206
Here, IRR(d,t) is the solar irradiation in terms of the output power divided by rated 207
power of PV panels at day d time slot t; PV(i) is the installed capacity of PV panels in 208
consumer i’s home; be(i,d,t) and se(i,d,t) are the volume of electricity consumer i buys 209
from and sells to the grid at day d time slot t, respectively; ca(i,d,c,t) is the power 210
consumption of consumer i’s specific controllable appliance c at day d time slot t. 211
NCL(i,d,t) is the total power consumption of consumer i’s non-controllable appliances 212
at day d time slot t; BA(i) is the installed capacity of battery in consumer i’s home; 213
11
brin(i,d,t) and brout(i,d,t) are the charge and discharge rate of consumer i’s battery at 214
day d time slot t. CHAEFF and DISEFF are the charge and discharge efficiency. 215
𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏 is the dual variable for balance constraints. 216
Batteries cannot be charged exceeding its capacity or discharged after empty. All 217
these constraints for the batteries are described in equation (2-2) – (2-6), respectively. 218
Particularly, equation (2-6) means the batteries should not be charged and discharged 219
simultaneously, which is a logical constraint. 220
221
∑ [𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖,𝑑𝑑,𝑘𝑘) − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖,𝑑𝑑, 𝑡𝑡)]𝑡𝑡𝑘𝑘=1 ≥ 0, �𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)�,∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-2) 222
∑ [𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖,𝑑𝑑,𝑘𝑘) − 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖,𝑑𝑑, 𝑡𝑡)]𝑡𝑡𝑘𝑘=1 ≤ 1, �𝜆𝜆𝐵𝐵𝐵𝐵−𝑢𝑢𝑢𝑢𝑢𝑢𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)� ,∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-3) 223
0 ≤ 𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) ≤ 1,∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-4) 224
0 ≤ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖,𝑑𝑑, 𝑡𝑡) ≤ 1,∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-5) 225
𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖,𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖,𝑑𝑑, 𝑡𝑡) = 0, �𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡)� ,∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-6) 226
227
Each controllable appliance can only work once for one hour (which is the length 228
of one entire time interval) in the given interval, as shown in equations (2-7) – (2-9). 229
230
∑ 𝑐𝑐𝑐𝑐(𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡)𝑡𝑡 = 𝐼𝐼𝑃𝑃(𝑐𝑐), �𝜆𝜆𝐶𝐶𝐵𝐵−𝑢𝑢𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑐𝑐, )� ,∀𝑖𝑖,𝑑𝑑, 𝑐𝑐 231
(2-7) 232
𝑐𝑐𝑐𝑐(𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡) ≤ 𝐼𝐼𝑃𝑃(𝑐𝑐),∀𝐷𝐷𝑆𝑆𝐵𝐵(𝑖𝑖,𝑑𝑑, 𝑐𝑐) ≤ 𝑡𝑡 ≤ 𝐷𝐷𝑆𝑆𝑆𝑆(𝑖𝑖,𝑑𝑑, 𝑐𝑐) (2-8) 233
𝑐𝑐𝑐𝑐(𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡) ≡ 0,∀𝑡𝑡 < 𝐷𝐷𝑆𝑆𝐵𝐵(𝑖𝑖,𝑑𝑑, 𝑐𝑐) 𝑏𝑏𝑏𝑏 𝑡𝑡 > 𝐷𝐷𝑆𝑆𝑆𝑆(𝑖𝑖,𝑑𝑑, 𝑐𝑐) (2-9) 234
12
235
Here, RP(c) is the rated power of controllable appliance c, 𝜆𝜆𝐶𝐶𝐵𝐵−𝑢𝑢𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑐𝑐, ) is 236
the dual variable for controllable appliance’s power consumption constraint. 237
𝐷𝐷𝑆𝑆𝐵𝐵(𝑖𝑖,𝑑𝑑, 𝑐𝑐) is the starting time and 𝐷𝐷𝑆𝑆𝑆𝑆(𝑖𝑖,𝑑𝑑, 𝑐𝑐) is the corresponding stop time of the 238
time interval in which the operation time of controllable appliance c can be shifted. 239
Consumers are only allowed to sell electric power generated from the distributed 240
PV to the grid because the chance of arbitrage should be avoided. In detail, the 241
possibility of buying electricity from the grid when the price is low and storing the 242
electricity in the battery, then selling it by discharging when the price is high should be 243
eliminated, which is shown in equation (2-10). On the other hand, logically, the 244
consumers should not purchase electricity power from and sell electricity power to the 245
retailer simultaneously. This is shown in equation (2-11). 246
247
0 ≤ 𝑠𝑠𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) ≤ 𝐼𝐼𝐼𝐼𝐼𝐼(𝑑𝑑, 𝑡𝑡) × 𝑃𝑃𝑃𝑃(𝑖𝑖),∀𝑖𝑖, 𝑑𝑑 (2-10) 248
𝑠𝑠𝑏𝑏(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) = 0, �𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡)� ,∀𝑖𝑖, 𝑑𝑑 (2-11) 249
250
2.3. Retailer pricing module 251
The obligation of the retailer is to meet electricity demand from all consumers. 252
Retailers purchase electricity from a variety of sources in a wholesale electricity market, 253
in which the electricity is generated by a variety of technologies and fuels. Each 254
technology has a different marginal cost, which is defined as the incremental cost 255
13
incurred to produce an additional unit of electric power. Naturally, (unless otherwise 256
compelled) the retailer will purchase electricity from the cheapest source first, leading 257
ultimately to a non-decreasing marginal cost curve. Therefore, if the retailer wants to 258
earn a fixed ratio profit, it is reasonable to approximately assume that the supply-price 259
curve is non-decreasing. 260
In this paper, an average-cost based pricing scheme is used as shown by equation 261
(2-12). Here, the NRL(d,t) is the non-residential electricity power load at day d time slot 262
t, which is set to be an exogenous constant, and the α and β are coefficients. 263
264
𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡) = 𝛼𝛼 ∑ [𝑏𝑏𝑏𝑏(𝑙𝑙,𝑑𝑑,𝑡𝑡)−𝑠𝑠𝑏𝑏(𝑙𝑙,𝑑𝑑,𝑡𝑡)]𝑖𝑖 +𝐶𝐶𝐶𝐶𝑁𝑁(𝑑𝑑,𝑡𝑡)�∑ ∑ 𝑁𝑁𝑁𝑁𝑁𝑁(𝑖𝑖,𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +∑ ∑ 𝑅𝑅𝑅𝑅(𝑖𝑖,𝑑𝑑,𝑐𝑐)𝑐𝑐𝑖𝑖 −∑ ∑ 𝑅𝑅𝑃𝑃(𝑖𝑖)×𝐼𝐼𝑅𝑅𝑅𝑅(𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +𝑁𝑁𝑅𝑅𝑁𝑁(𝑑𝑑,𝑡𝑡)�
24
+265
𝛽𝛽, ( 𝜆𝜆𝑢𝑢𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏(𝑑𝑑, 𝑡𝑡) ),∀𝑑𝑑, 𝑡𝑡 (2-12) 266
Such a pricing function can ensure the supply-price curve is non-decreasing. 267
Furthermore, the pricing mechanism also encourages users to schedule their energy 268
consumption and batteries in a way that the energy demand is more equally distributed 269
over all time slots. As shown in the equation, α + β is the average price. 270
2.4. Non-cooperation game theoretical complementarity model 271
The objective of each residential consumer is to minimize his/her energy 272
expenditure on electricity consumption in a whole year as is shown in formula (2-13). 273
Each consumer decides his/her own load commitment and battery operation pattern. All 274
information is public and there is no collusion between consumers. Therefore this is a 275
typical non-cooperation game. 276
14
277
min𝑏𝑏𝑏𝑏,𝑠𝑠𝑏𝑏,𝑏𝑏𝑐𝑐,𝑢𝑢𝑝𝑝
{∑ [∑ 𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡)𝑡𝑡 − ∑ 𝐶𝐶𝐼𝐼𝑆𝑆 × 𝑃𝑃𝑃𝑃(𝑖𝑖) × 𝐼𝐼𝐼𝐼𝐼𝐼(𝑑𝑑, 𝑡𝑡)𝑡𝑡 ] ×𝑑𝑑278
𝐷𝐷𝐵𝐵𝐷𝐷𝐷𝐷(𝑑𝑑)} ,∀𝑖𝑖 (2-13) 279
280
In the objective function (2-13), FIT is the feed-in tariff rate and DAYS(d) is the 281
amount of the representative day, d, in a year. The optimization is subject to constraints 282
(2-1) – (2-12). The Karush–Kuhn–Tucker conditions (also known as KKT or first order 283
conditions) of the non-cooperation game model are established as (2-14) – (2-19). The 284
objective function is quadratic and all the constraints are linear, and thus the KKT 285
conditions are necessary and sufficient for the optimization of the objective function. 286
Since the non-cooperation game model is a simultaneous-move game, solving the KKT 287
conditions simultaneously for expenditure minimizing problems yields a Nash 288
equilibrium solution. [24] 289
290
𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡) × 𝐷𝐷𝐵𝐵𝐷𝐷𝐷𝐷(𝑑𝑑) + 𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) + 𝜆𝜆𝑢𝑢𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏(𝑑𝑑, 𝑡𝑡) ×291
𝛼𝛼×𝑏𝑏𝑏𝑏(𝑙𝑙,𝑑𝑑,𝑡𝑡)[∑ ∑ 𝐶𝐶𝐶𝐶𝑁𝑁(𝑙𝑙,𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +∑ ∑ 𝐶𝐶𝑅𝑅(𝑙𝑙,𝑑𝑑,𝑐𝑐)𝑐𝑐𝑖𝑖 −∑ ∑ 𝑅𝑅𝑃𝑃(𝑙𝑙)×𝐼𝐼𝐶𝐶𝐶𝐶(𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +𝐶𝐶𝐶𝐶𝑁𝑁(𝑑𝑑,𝑡𝑡)]/24
+292
𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) × 𝑠𝑠𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) ⊥ 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡),∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-14) 293
294
−𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) − 𝜆𝜆𝑢𝑢𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏(𝑑𝑑, 𝑡𝑡) × 𝛼𝛼×𝑠𝑠𝑏𝑏(𝑙𝑙,𝑑𝑑,𝑡𝑡)�∑ ∑ 𝑁𝑁𝑁𝑁𝑁𝑁(𝑖𝑖,𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +∑ ∑ 𝑅𝑅𝑅𝑅(𝑖𝑖,𝑑𝑑,𝑐𝑐)𝑐𝑐𝑖𝑖 −∑ ∑ 𝑅𝑅𝑃𝑃(𝑖𝑖)×𝐼𝐼𝑅𝑅𝑅𝑅(𝑑𝑑,𝑡𝑡)𝑖𝑖𝑡𝑡 +𝑁𝑁𝑅𝑅𝑁𝑁(𝑑𝑑,𝑡𝑡)�
24
+295
𝜆𝜆𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) ⊥ 𝑠𝑠𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡),∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 (2-15) 296
297
15
−𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) + 𝜆𝜆𝐶𝐶𝐵𝐵−𝑢𝑢𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑐𝑐) ⊥ 𝑐𝑐𝑐𝑐(𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡),∀𝑖𝑖,𝑑𝑑, 𝑐𝑐, 𝑡𝑡 298
(2-16) 299
300
−𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏 × 𝐵𝐵𝐵𝐵(𝑖𝑖) × 𝑁𝑁𝐶𝐶𝐵𝐵𝐶𝐶𝐶𝐶𝐶𝐶 + ∑ 𝜆𝜆𝐵𝐵𝐵𝐵−𝑢𝑢𝑢𝑢𝑢𝑢𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)𝑡𝑡𝑘𝑘=0 −301
∑ 𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)𝑡𝑡𝑘𝑘=0 + 𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) ⊥ 𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖,𝑑𝑑, 𝑡𝑡),∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 302
(2-17) 303
304
−𝜆𝜆𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑐𝑐𝑏𝑏 × 𝐵𝐵𝐵𝐵(𝑖𝑖) × 𝑁𝑁𝐶𝐶𝐵𝐵𝐶𝐶𝐶𝐶𝐶𝐶 − ∑ 𝜆𝜆𝐵𝐵𝐵𝐵−𝑢𝑢𝑢𝑢𝑢𝑢𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)𝑡𝑡𝑘𝑘=0 +305
∑ 𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑏𝑏𝑙𝑙(𝑖𝑖,𝑑𝑑, 𝑡𝑡)𝑡𝑡𝑘𝑘=0 + 𝜆𝜆𝐵𝐵𝐵𝐵−𝑏𝑏𝑙𝑙𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏𝑏𝑏(𝑖𝑖, 𝑑𝑑, 𝑡𝑡) × 𝑏𝑏𝑏𝑏𝑖𝑖𝑖𝑖(𝑖𝑖,𝑑𝑑, 𝑡𝑡) ⊥ 𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑏𝑡𝑡(𝑖𝑖,𝑑𝑑, 𝑡𝑡),∀𝑖𝑖,𝑑𝑑, 𝑡𝑡 306
(2-18) 307
308
∑ 𝑏𝑏𝑏𝑏(𝑖𝑖,𝑑𝑑, 𝑡𝑡) × 𝐷𝐷𝐵𝐵𝐷𝐷𝐷𝐷(𝑑𝑑)𝑙𝑙 + 𝜆𝜆𝑢𝑢𝑙𝑙𝑙𝑙𝑐𝑐𝑏𝑏(𝑑𝑑, 𝑡𝑡) ⊥ 𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡),∀𝑑𝑑, 𝑡𝑡 (2-19) 309
310
2.5. Tool 311
The non-cooperation game theoretical complementarity model is developed in the 312
General Algebraic Modeling System (GAMS) [25] as a mixed complementarity 313
problem (MCP). The mathematical properties of existence and uniqueness for the MCP 314
solution can be found in [26, 27]. In the present study, the MCP problem is solved using 315
the PATH solver [28]. It takes 68s to solve this problem on a computer with i7 2.5 GHZ 316
CPU and 4G memory. 317
GAMS is a platform that uses algebraic language and efficient solvers for analyzing 318
16
complex and large-scale linear, nonlinear, integer, and complementarity problems. The 319
PATH solver is a Newton-based algorithm for solving complementarity problems. 320
3. Case study 321
The developed non-cooperation game theoretical power market complementarity 322
model was applied to a district in China. All the residential consumers participate in the 323
real-time pricing program. To investigate the promotion impact of smart levels of 324
residential consumers on the acceptability of distributed PV power, the economic 325
performance of installed PV power, consumers’ expenditure on power consumption and 326
the optimal operation pattern of electric appliances under the non-cooperation game 327
will be analyzed using the proposed model. 328
3.1. Data 329
The consumption of electricity has obvious time-dependent characteristics. 330
Residential consumers always arrange their electric appliances’ operations according 331
to their life style arbitrarily when demand response is not involved. Seasonal difference 332
is also a factor in determining the load pattern. In summer and winter, the load is 333
relatively higher than that in spring and autumn due to the high electricity demand for 334
cooling and heating. Therefore, three representative days for summer, winter, and mid-335
seasons respectively are selected, upon which to base the assessment of the whole year. 336
17
3.1.1. Non-residential load 337
Non-residential load data was obtained using a foreign city as reference [6]. In 338
China, non-residential load accounts for 80% of the total load [29]. The non-residential 339
consumers don’t respond to real-time price as described in Section 2, therefore their 340
load is constant, as is shown in Figure 2. The peak load appears at noon and the lowest 341
load appears before dawn. The power load in summer and winter is higher than that in 342
a typical mid-seasonal day. 343
3.1.2. Residential load and variable time zone 344
Taking reference [30] as reference basis, we conducted a survey on the residential 345
consumers’ life style with regards to electricity consumption. Eleven households were 346
sampled, and their electric appliances’ daily operation patterns were recorded. 347
According to the survey result, we summarized the residential load data. The residential 348
load consists of two parts. The first part is the non-controllable load, which consists of 349
all the non-controllable appliances as shown in Figure 3. The second part is the 350
controllable load, which consists of all the controllable appliances, and the initial 351
distribution of the controllable load is depicted in Figure 4. 352
As is mentioned in Section 1, the rapid upgrading of smart house technologies leads 353
to different demand response abilities (smart levels) in houses constructed in different 354
technical eras. In the present study, five types of residential consumers with different 355
abilities to respond to price changes have been taken into consideration. The smart 356
18
house technologies available to them in order of their houses’ age are: (1) automated 357
controller and access to real-time price data; (2) remote controller and access to real-358
time price data; (3) timing start and access to real-time price data; (4) only access to 359
real-time price; and (5) no access to even real-time price, in reverse order of increased 360
smart level respectively. To quantify their different smart levels, it is assumed that the 361
length of the time interval during which energy consumption can be shifted is 24 hours 362
for consumer 1, 11 hours for the consumer 2, 5 hours for the consumer 3, 3 hours for 363
the consumer 4, and 1 hour for the consumer 5, respectively. 364
In the present study, we take consumer 3 as an example to illustrate the meaning 365
of the time interval length: if resident 3 is accustomed to using the washing machine at 366
1:00 pm before going out to work in the afternoon, he can choose to shift the operation 367
within the time interval 11:00 am to 3:00 pm, of which 1:00 pm is the midpoint. In 368
particular, consumer 1 can shift his load to any other hour within the day and consumer 369
5 cannot shift any load. 370
Each type of residential consumer includes 1000 households. The 5000 households 371
share the same non-controllable load and initial controllable load. 372
3.1.3. Solar irradiation 373
The solar irradiation corresponding to the three representative days is depicted in 374
Figure 5. The amount of PV generation power can be calculated by equation (3-1). [6, 375
16] 376
377
19
𝑃𝑃𝑅𝑅𝑃𝑃 = 𝜂𝜂𝑅𝑅𝑃𝑃 × 𝑃𝑃𝑃𝑃 × 𝐼𝐼𝑅𝑅𝑃𝑃𝐼𝐼 × �1 − 0.005 × (𝑡𝑡𝐶𝐶𝐶𝐶 − 25)� (3-1) 378
379
where, 𝜂𝜂𝑅𝑅𝑃𝑃 is the conversion efficiency of the solar cell array (14.4%), PV is the 380
rated capacity of PV panels, 𝐼𝐼𝑅𝑅𝑃𝑃𝐼𝐼 is the solar irradiation on an inclined surface (kW/m2), 381
and 𝑡𝑡𝐶𝐶𝐶𝐶 is the outside air temperature (° C). The angle of incidence of the solar array 382
panel is 30 degrees. Thus, the irradiation coefficient IRR can be calculated by equation 383
(3-2). 384
385
𝐼𝐼𝐼𝐼𝐼𝐼 = 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃� = 𝜂𝜂𝑃𝑃𝑃𝑃 × 𝐼𝐼𝑃𝑃𝑃𝑃𝐼𝐼 × �1− 0.005 × (𝑡𝑡𝑁𝑁𝐼𝐼 − 25)� (3-2) 386
387
3.2. Policy assumptions 388
In this paper, the subsidy policy for distributed PV panels is assumed to be a fixed 389
subsidy on every kWh power generated by PV panels [1, 31, 32], which means 390
residential consumers cannot be paid via sending excess PV power to the grid, resulting 391
in incentive for consumers to utilize the PV power as much as possible. 392
3.3. Simulation study 393
The battery capacities and PV capacities are assumed to be the same for all 394
consumers. The battery capacities vary from 0 to 3 kWh in steps of 1kWh. The PV 395
capacities vary from 0 to 3.5 kW in step of 0.5 kW. Different combinations of PV 396
capacities and battery capacities lead to different optimized operation and economic 397
20
benefit results. The numerical results will be provided in Section 4. 398
4. Result 399
4.1. Electricity expense 400
The simulations were run for all the combinations of PV capacities and battery 401
capacities. The annual total expenditures of each type of consumers on electricity power 402
consumption have been summarized in Figure 6 in the form of the annual expenses 403
saved comparing to consumer 5, whose smart level is the lowest. The saved money is 404
the value of the “smart level”. As shown in the results, consumers with higher smart 405
levels can save more money. With the PV installed capacity increase, the superiorities 406
become even more significant. When there is a 3.5 kW PV but no battery, a consumer 407
with the highest smart level will spend 36.6% less expenditure than the consumer with 408
the lowest smart level annually. However, when same capacity battery is integrated, the 409
superiorities in saving electricity expenditure of the consumers with higher smart level 410
are attenuated compared to low smart level ones. For example, with 2 kWh battery and 411
3.5 kW PV, a consumer with the highest smart level can save only 25.2% compared to 412
the lowest smart level consumer. This indicates that the batteries can reduce the 413
economic performance gap between consumers with different smart levels. The reasons 414
for this are investigated as described in section 4.3. 415
21
4.2. Marginal revenue of PV power 416
The cost of PV panels and batteries have not been counted into the expenses in 417
this study, because we only focus on the promotion impacts of the consumers’ smart 418
levels on the integration of PV power. Moreover, the cost of PV panels and battery 419
modules are reducing dramatically year by year [33, 34]. Therefore, the marginal 420
revenue was considered a more reasonable index than levelised cost of electricity. 421
Marginal revenue is the increment when capacity increases one unit. In practice, if the 422
capital cost of a certain capacity of PV is less than its marginal revenue, it is 423
economically acceptable. 424
As shown in Figure 7, the consumers with higher smart levels (consumer 1 and 425
consumer 2) have flatter marginal revenue for new installed PV. On the other hand, the 426
consumers with lower smart levels (consumer 4 and consumer 5) have sharply dropping 427
marginal revenue. For consumer 5, the marginal revenue of each 0.5 kW PV panel 428
decreased from 404 RMB for the first unit PV panel, to 85 RMB for the seventh PV 429
panel. However, for consumer 1, the marginal revenue of each 0.5 kW PV panel only 430
decreased from 399 RMB for the first unit PV panel, to 341 RMB for the seventh unit 431
PV panel. The difference indicates that the higher the smart level of the consumers, the 432
more PV panels can be economically acceptable. Furthermore, batteries can help the 433
consumers with lower smart levels to delay the decrease of PV marginal revenue 434
effectively. Taking consumer 3 as an example, the marginal revenue of PV starts to 435
decrease sharply when PV capacity is 1.5 kW when there is no battery; and the turning 436
22
point of marginal revenue of PV moves to 2kW PV capacity when there is a 1 kWh 437
battery. Furthermore, the turning point will move to 2.5kW and 3 kW PV capacity 438
respectively when 2kWh and 3kWh batteries are deployed. 439
On the other hand, according to the numerical result shown in Figure 7, compared 440
to consumers 3, 4 and 5, the battery has no obvious impacts on the marginal revenue of 441
PV for consumer 1 and consumer 2 whose smart level is relatively higher. Therefore, 442
consumers with a high smart level can economically accept a large generating capacity 443
of PV power, and thus the battery contributes little in further promoting distributed PV 444
installation for them. 445
4.3. Price variance under different situations 446
The daily price variance can be calculated by equation (4-1). 447
448
𝑝𝑝𝑏𝑏𝑖𝑖𝑐𝑐𝑏𝑏 𝑣𝑣𝑐𝑐𝑏𝑏𝑖𝑖𝑐𝑐𝑖𝑖𝑣𝑣𝑏𝑏 = 124−1
∑ �𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡) − 124∑ 𝑝𝑝𝑏𝑏(𝑑𝑑, 𝑡𝑡)𝑡𝑡 �
2𝑡𝑡 (4-1) 449
450
The price variances under different situations are exhibited in Figure 8. It is 451
depicted that the price variance decreases with the increase of battery capacity. The 452
consumers with high smart levels are able to utilize the price fluctuation to save money 453
by purchasing electricity power during lower price periods. Since the total load is the 454
same for all the consumers, the different smart levels do not cause any difference when 455
the price variances become zero (price is a constant). In other words, smarter consumers 456
can benefit from higher price variance more. 457
23
Analyzing the results shown in Figure 7 and Figure 8 together indicates that 458
batteries can help consumers with relatively lower smart levels to promote PV panel 459
installation. When the battery capacities increases from zero to 1 kWh, consumer 1 can 460
save 31 RMB with a 3.5 kW PV panel installed, while consumer 5 can save 281 RMB. 461
The reason is that the battery can reduce price variances effectively, and the consumers 462
with lower smart levels can benefit more when the price variance becomes low. 463
This, following-on from the previous section has significant implications for policy 464
and the roll-out of PV and its supporting technologies. In the case of relatively smart 465
technologies, batteries are not so crucial, while dealing with less-smart demand-side 466
infrastructure encourages battery installation. However, the capital and maintenance 467
costs of these alternative installations should also be carefully considered with regards 468
to appropriate subsidisation and investment. 469
4.4. Operation pattern 470
When the non-cooperative game reaches an equilibrium state, every consumer’s 471
appliance operation pattern is considered to be optimal. The equilibrium was obtained 472
from the hour-by-hour simulation and the feasibility was inherent. A typical day in 473
summer with 1kWh battery, 3kW PV was chosen as an example. The operation patterns 474
of consumer 1 and consumer 5 are shown in Figure 9. It is apparent that consumer 1 475
responds to the real-time price by using more power when the price is lower before 476
dawn and using less power when price is higher in the evening. 477
24
5. Conclusion 478
In the present study, a non-cooperation game theoretical power market 479
complementarity model was developed to study the equilibrium real-time power price 480
for residential consumers considering smart residential micro-grid and distributed PV 481
panels. The model was applied to case studies to uncover the impacts of smart levels 482
on the economic integration of distributed PV power, and the impacts of PV capacities 483
and battery capacities on consumers’ power expenses. 484
Five types of residential consumers with different capabilities of responding to 485
price changes were considered to be playing in the game and 32 different combinations 486
of battery capacities and PV capacities were simulated. The results show that: (a) 487
residential consumers with higher smart levels at higher integration of PV are able to 488
benefit more from the non-cooperative game in form of saving money on electricity 489
expenditure. The preponderances increase with the increase in PV capacity and 490
decrease with the increase of battery capacity. In this study, consumers with the highest 491
smart level can save 36.6% on electricity expenditure compared to the consumers with 492
the lowest smart level at most; (b) the marginal revenue of new installed PV panels 493
decreases for all residential consumers. However, consumers with lower smart level 494
suffer a sharper drop. The difference of the marginal revenue among different users can 495
reach 300% in this study. It can be concluded that consumers with higher smart levels 496
are able to economically integrate a larger PV capacity. (c) Batteries can help consumers 497
25
with relatively low smart levels to mitigate the decrease of PV marginal revenue, 498
however, they cannot significantly raise the PV marginal revenue for consumers with 499
high smart levels. 500
The real-time power price was obtained simultaneously. By analyzing the price, it 501
is uncovered that consumers with lower smart levels get more benefit from the increase 502
of battery capacity as a result of the decrease of price variances. When there is no PV, 503
the price variance in a typical summer day as an example decreases by 94.4% with the 504
integrated battery capacity increasing from 0 to 3 kWh and the electricity consumption 505
difference between consumers with highest and lowest smart level decreases by 50%. 506
507
26
Figures 508
Figure 1 Structure of the target system 509
Figure 2 Non-residential electricity load 510
Figure 3 Non-controllable load of one household 511
Figure 4 Initial controllable load of one household 512
Figure 5 Solar irradiation coefficient in different seasons 513
Figure 6 Saved annual electricity power expense of one household comparing to 514
consumer 5 under different PV and battery capacity 515
Figure 7 Marginal revenue of new installed PV capacity under different battery 516
capacity 517
Figure 8 Daily variance of real-time power price under different PV and battery 518
capacity 519
Figure 9 Operation patterns of consumer 1 and consumer 5 in the summer 520
represent day with 1 kWh battery and 3 kW PV 521
522
523
27
524
Figure 1 Structure of the target system 525
526
527
Figure 2 Non-residential electricity load 528
529
28
530
Figure 3 Non-controllable load of one household 531
532
533
Figure 4 Initial controllable load of one household 534
535
29
536
Figure 5 Solar irradiation coefficient in different seasons 537
538
Figure 6 Saved annual electricity power expense of one household comparing to 539
consumer 5 under different PV and battery capacity 540
541
30
Figure 7 Marginal revenue of new installed PV capacity under different battery 542
capacity 543
544
Figure 8 Daily variance of real-time power price under different PV and battery 545
31
capacities 546
547
548
Figure 9 Operation patterns of consumer 1 and consumer 5 in the summer represent 549
day with 1 kWh battery and 3 kW PV 550
551
Acknowledgements 552
The study was supported by the Science Foundation of China University of Petroleum ,553
Beijing (No. 2462013YJRC015). 554
555
32
Reference 556
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